AP Calculus AB Unit 2 Notes: Differentiating Products and Quotients

Product rule

A differentiation rule for a product of two differentiable functions: if h(x)=f(x)g(x), then h'(x)=f'(x)g(x)+f(x)g'(x).

Derivative of a product (correct structure)

For f(x)g(x), add two contributions: (derivative of first)(second) + (first)(derivative of second).

Instantaneous rate of change

What a derivative represents: the rate at which a quantity changes at a specific input value (as $Δx \to 0$).

Common product-rule mistake: f'(x)g'(x)

An incorrect guess that the derivative of f(x)g(x) equals f'(x)g'(x); it generally fails because both factors’ changes contribute additively, not multiplicatively.

Small-change expansion for a product

Using Δ(fg)=\big[(f + Δf)(g + Δg)\big] - fg to get Δ(fg)=fΔg + gΔf + ΔfΔg, which motivates the product rule.

“Extra small” term $\frac{ΔfΔg}{Δx}$

In the product-rule derivation, the term $\frac{ΔfΔg}{Δx}$ goes to 0 as $Δx \to 0$ because both $Δf$ and $Δg$ become very small.

Product rule application step: identify factors

When differentiating a product, first label the two factors as f(x) and g(x) (or group multiple factors into two parts).

Product rule application step: differentiate separately

Compute f'(x) and g'(x) before substituting into f'(x)g(x)+f(x)g'(x).

Product rule application step: simplify last

Only simplify after writing the correct product-rule structure, to avoid algebra errors.

Mnemonic: “Left d right + right d left”

Memory aid for product rule: keep the left factor and differentiate the right, then add keep the right and differentiate the left.

Misconception: differentiation distributes over multiplication

The false idea that \frac{d}{dx}[f(x)g(x)] can be found by “distributing” the derivative across multiplication; this works for addition, not multiplication.

Non-like terms (sin and cos)

Expressions like sin(x) and cos(x) are not like terms and generally cannot be combined by addition/subtraction simplification.

Product of more than two factors (grouping)

To differentiate p(x)q(x)r(x), group into two factors (e.g., (p(x)q(x))r(x)) and apply the product rule, possibly more than once.

Quotient rule

A differentiation rule for a ratio: if h(x)=f(x)/g(x) with g(x)
eq 0, then h'(x)=\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}.

Derivative of a quotient (order matters)

In \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}, the subtraction is not symmetric; reversing the order changes the sign of the answer.

Common quotient-rule mistake: f'(x)/g'(x)

An incorrect idea that \frac{d}{dx}[f(x)/g(x)] equals \frac{f'(x)}{g'(x)}; this is not a valid differentiation rule.

“Per” quantity interpretation

Quotients often model “per” measurements (e.g., cost per item); the quotient rule accounts for changes in both numerator and denominator.

Quotient as a product with negative exponent

Rewrite \frac{f(x)}{g(x)} as f(x)(g(x))^{-1} to connect the quotient rule to the product rule (and chain rule).

Derivative of (g(x))^{-1}

Using chain rule: \frac{d}{dx}\big[(g(x))^{-1}\big] = -(g(x))^{-2}\bullet g'(x), which helps derive the quotient rule.

Why the denominator is squared in the quotient rule

The (g(x))^2 arises when combining terms over a common denominator after differentiating f(x)(g(x))^{-1}.

Mnemonic: “Low d high minus high d low, over low squared”

Memory aid for quotient rule: \frac{(denominator\bullet derivative ext{ of numerator}) - (numerator\bullet derivative ext{ of denominator})}{(denominator)^2}.

Quotient-rule trap: forgetting parentheses

A common error where only part of the numerator is subtracted (e.g., missing parentheses around f(x)g'(x)), leading to sign mistakes.

Quotient-rule trap: not squaring the entire denominator

An error where the denominator is written as g(x) instead of (g(x))^2, or only part of g(x) is squared (e.g., misreading (x-1)^2).

Legal vs. illegal cancellation before differentiating

You may cancel common factors (when valid and mindful of domain), but you cannot cancel terms across addition/subtraction, and cancellation can change domain if not handled carefully.

When to avoid the quotient rule by rewriting

Sometimes rewrite $\frac{1}{x^n}$ as $x^{-n}$ (or $\frac{f(x)}{x^n}$ as $f(x)x^{-n}$) to use power/product rules; be cautious if the denominator is a complicated expression where chain rule becomes error-prone.